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MaxMinMax problem and sparse equations over finite fields

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Abstract

Asymptotical complexity of polynomial equation systems over finite field \(F_q\) is studied. Let \({\mathcal {X}}=\{X_1,\ldots ,X_m\},\, |\bigcup _{i=1}^{m}X_i|\le n\) be a fixed family of variable sets and the polynomials \(f_i(X_i)\) are taken independently and uniformly at random from the set of all polynomials of degree \({\le }q-1\) in each of the variables in \(X_i\). In particular, it is proved if \(|X_i|\le 3, m=n\), then the average complexity of finding all solutions in \(F_q\) to \(f_i(X_i)=0\, (1\le i\le m)\) is at most \( q^{\frac{n}{5.7883}+O(\log n)}\) for arbitrary \({\mathcal {X}}\) and \(q\). The proof is based on a detailed analysis of MaxMinMax problem, a novel problem for hypergraphs.

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Acknowledgments

I am grateful to Peter Horak for a number of suggestions on improving the presentation of an earlier variant of this work. The competition with [14] stimulated my research. I am grateful to four anonymous referees for their comments.

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  1. Department of Informatics, University of Bergen, Bergen, Norway

    Igor Semaev

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  1. Igor Semaev
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Correspondence to Igor Semaev.

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Communicated by I. Shparlinski.

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Semaev, I. MaxMinMax problem and sparse equations over finite fields. Des. Codes Cryptogr. 79, 383–404 (2016). https://doi.org/10.1007/s10623-015-0058-6

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